Solving the Maximum Cardinality Bin Packing Problem with a Weight Annealing-Based Algorithm

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Solving the Maximum Cardinality Bin Packing Problem with a Weight Annealing-Based Algorithm . Kok-Hua Loh University of Maryland Bruce Golden University of Maryland Edward Wasil American University. 10 th ICS Conference January 2007. Outline of Presentation. Introduction

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### Solving the Maximum Cardinality Bin Packing Problem with a Weight Annealing-Based Algorithm

Kok-Hua Loh

University of Maryland

Bruce Golden

University of Maryland

Edward Wasil

American University

10th ICS Conference

January 2007

Outline of Presentation
• Introduction
• Concept of Weight Annealing
• Maximum Cardinality Bin Packing Problem
• Conclusions

1

WeightAnnealing Concept
• Assigning different weights to different parts of a combinatorial problem to guide computational effort to poorly solved regions.
• Ninio and Schneider (2005)
• Elidan et al. (2002)
• Allowing both uphill and downhill moves to escape from a poor local optimum.
• Tracking changes in objective function value, as well as how well every region is being solved.
• Applied to the Traveling Salesman Problem. (Ninio and Schneider 2005)
• Weight annealing led to mostly better results than simulated annealing.

2

One-Dimensional Bin Packing Problem (1BP)
• Pack a set of N = {1, 2, …, n} items, each with size ti, i=1, 2,…, n, into identical bins, each with capacity C.
• Minimize the number of bins without violating the capacity constraints.
• Large literature on solving this NP-hard problem.

3

Outline of Weight Annealing Algorithm
• Construct an initial solution using first-fit decreasing.
• Compute and assign weights to items to distort sizes according to the packing solutions of individual bins.
• Perform local search by swapping items between

all pairs of bins.

• Carry out re-weighting based on the result of the previous optimization run.
• Reduce weight distortion according to a cooling schedule.

4

Neighborhood Search for Bin Packing Problem
• From a current solution, obtain the next solution by swapping items between bins with the following objective function (suggested by Fleszar and Hindi 2002).

5

Neighborhood Search for Bin Packing Problem
• Swap schemes
• Swap items between two bins.
• Carry out Swap (1,0), Swap (1,1), Swap (1,2), Swap (2,2) for all pairs of bins.
• Analogous to 2-Opt and 3-Opt.
• Swap (1,0) (suggested by Fleszar and Hindi 2002)

Bin α

Bin β

Bin α

Bin β

• Need to evaluate only the change in the objective function value.

6

Neighborhood Search for Bin Packing Problem
• Swap (1,1)

( fnew = 164)

(f = 162)

• Swap (1,2)

( fnew = 164)

(f = 162)

7

Weight Annealing for Bin PackingProblem
• Weight of item i

wi = 1 + K ri

• An item in a not-so-well-packed bin, with large ri,

will haveits size distorted by a large amount.

• No size distortions for items in fully packed bins.
• K controls the size distortion, given a fixedri .

8

Weight Annealing for Bin Packing Problem
• Weight annealing allows downhill moves in a maximization

problem.

• Example C = 200, K= 0.5,

Transformed space f = 70126.3

Original space f = 63325

Transformed space f new = 70132.2

Original space f new= 63125

• Transformed space -uphill move
• Original space - downhill move

9

Maximum Cardinality Bin Packing Problem (MCBP)
• Problem statement
• Assign a subset of n items with sizes ti to a fixed number of m bins

of identical capacity c.

• Maximize the number of items assigned.
• Formulation

10

Maximal Cardinality Bin Packing Problem
• Practical applications
• Computing.
• Assign variable-length records to a fixed amount of storage.
• Maximize the number of records stored in fast memory so as to ensure a minimum access time to the records.
• Management of real time multi-processors.
• Maximize the number of completed tasks with varying job durations before a given deadline.
• Computer design.
• Designing processors for mainframe computers.
• Designing the layout of electronic circuits.

11

Bounds for Maximal Cardinality Bin Packing Problem
• We use the three upper bounds on the optimal number of items developed by Labbé, Laporte, and Martello (2003):
• We use the two lower bounds on the minimal number of bins developed by Martello and Toth (1990): L2 and L3.

12

Outline of Weight Annealing Algorithm (WAMC)
• Arrange items in the order of non-decreasing size.
• Compute a priori upper bound on the optimal number of items (U*).
• U* =
• Update ordered list by removing item i with size ti for which i >U*.

(The optimal solution z* is obtained by selecting the first z* smallest items.)

• Improve the upper bound U*.
• Find lower bound on the minimum number of bins required (L3).
• If L3 > m, reduce U* by 1.
• Update ordered list by removing item i with size ti for which i >U*.
• Iterate until L3 = m.
• Find feasible packing solution for the ordered list with the weight annealing algorithm for 1BP.
• Output results.

13

Solution Procedures for MCBP
• Enumeration algorithm (LA) by Labbé, Laporte, and Martello (2003)
• Compute a priori upper bounds.
• Embed the upper bounds into an enumeration algorithm.
• Branch-and-price algorithm (BP) by Peeters and Degraeve (2006)
• Compute a priori and LP upper bounds.
• Solve the problem with heuristics in a branch-and-price framework.

14

Test Problems
• Labbé, Laporte, and Martello (2003)
• 180 combinations of three parameters
• number of bins m = 2, 3, 5, 10, 15, 20
• capacity c = 100, 120, 150, 200, 300, 400, 500, 600, 700, 800
• size interval [tmin, 99]; tmin = 1, 20, 50
• For each combination (m, c, tmin), create 10 instances by generating item size ti in the given size interval until
• Peeters and Degraeve (2006)
• 270 combinations of three parameters
• capacity c=1000, 1200, 1500, 2000, 3000, 4000, 5000, 6000, 7000, 8000
• size interval [tmin, 999]; tmin = 1, 20, 50
• desired number of items
• For each combination (m, c, tmin), create 10 instances by generating item size ti in the given size interval until

15

Computational Results
• Results on the test problems of Labbé, Laporte, and Martello

(2003)

• Generated 1800 problems for testing on WAMC .
• LA and BP used a different set of 1800 problems.
• Number of instances solved to optimality
• BP 1800
• WAMC 1793
• LA 1759
• Average running times
• BP < 0.01 sec (500 MHz Intel Pentium III )
• WAMC 0.03 sec (3 GHz Intel Pentium IV )
• LA 3.16 sec (Digital VaxStation 3100)

16

Computational Results
• Generated 2700 problems for testing on WAMC; BP used a

different set of 2700 problems.

• Computational Results
• WAMC outperforms BP.
• BP had difficulties solving instances with
• Large bin capacities (5000-8000)
• Large number of items (350-500).
• WAMC solved all instances with bin capacities
• WAMC was faster.

17

Conclusions
• WAMC is easy to understand and simple to code.
• Weight annealing has wide applicability(1BP, 2BP).
• WAMC produced high-quality solutions to the maximum cardinality bin packing problem.
• WAMC solved 99% (4458/4500) of the test instances to optimality with an average time of a few tenths of a second.

18